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Nankai Tracts in Mathematics - Vol. 1
SCISSORS CONGRUENCES, GROUP HOMOLOGY AND CHARACTERISTIC CLASSES
by Johan L Dupont (University of Aarhus, Denmark)
These lecture notes are based on a series of lectures given at the Nankai Institute of Mathematics in the fall of 1998. They provide an overview of the work of the author and the late Chih-Han Sah on various aspects of Hilbert's Third Problem: Are two Euclidean polyhedra with the same volume "scissors-congruent", i.e. can they be subdivided into finitely many pairwise congruent pieces? The book starts from the classical solution of this problem by M Dehn. But generalization to higher dimensions and other geometries quickly leads to a great variety of mathematical topics, such as homology of groups, algebraic K-theory, characteristic classes for flat bundles, and invariants for hyperbolic manifolds. Some of the material, particularly in the chapters on projective configurations, is published here for the first time.
Contents:
- Introduction and History
- Scissors Congruence Group and
Homology
- Homology of Flag Complexes
- Translational Scissors Congruences
- Euclidean Scissors Congruences
- Sydler's Theorem and Non-Commutative Differential Forms
- Spherical Scissors Congruences
- Hyperbolic Scissors Congruence
- Homology of Lie Groups Made Discrete
- Invariants
- Simplices in Spherical and Hyperbolic 3-Space
- Rigidity of Cheeger–Chern–Simons Invariants
- Projective Configurations and Homology of the Projective Linear Group
- Homology of Indecomposable Configurations
- The Case of PGl(3,F)
Readership: Graduate students and researchers in geometry and topology.
| 176pp |
Pub. date: Feb 2001 |
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